Upper bound on the black-hole critical exponents

We derive the model-independent upper bound γ≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \le 1$$\end{document} on the Choptuik critical exponents that characterize the scaling relation rBH∝(p-p∗)γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{\text {BH}}\propto (p-p^*)^{\gamma }$$\end{document} at the threshold of black-hole formation in type-II nearly self-similar gravitational collapse scenarios [here Δp≡(p-p∗)/p∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta p\equiv (p-p^*)/p^*$$\end{document} and rBH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{\text {BH}}$$\end{document} are, respectively, the initial phase space deviation of a near-critical initial data from the exact self-similar critical solution and the horizon radius of the corresponding super-critical dynamically formed black hole]. Our compact proof is based on the non-decreasing nature of the coarse-grained phase space volume of the dynamical system. The analytically derived upper bound is confirmed by the numerical results that have been published in the physics literature about the Choptuik black-hole critical phenomena during the last three decades. The present work may shed new light on the relation between the critical phenomena in gravitational collapse and statistical physics.